Finite Group Databasebeta

← C22×#15668040 #26034987

Subgroups — C22×#15668040 #26034987

Order 11392.

1720 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 565 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13083574C211yesmaximal, characteristic, complemented
2C23×D356C211yesmaximal, characteristic, complemented
3C2×#15668040C2124yesmaximal, direct product factor
4C22×#13083563C212yesmaximal, complemented
5C22×D712C212yesmaximal, complemented
6C22×M16×C89C211yesmaximal, characteristic, complemented
7C24×D89—21no—
8C2×#13083574—26no—
9C2×#13083574C22112yescomplemented
10C22×Dic178C2211yescharacteristic, complemented
11C22×D356—21no—
12C22×D356—26no—
13C22×D356C2212yes—
14C22×D356C22112yescomplemented
15C22×D356C2211yescharacteristic, complemented
16C22×C22⋊D89—21no—
17C22×C4×D89—21no—
18#15668040C22164yesdirect product factor
19C2×#13083563C22124yescomplemented
20C2×D712C22124yescomplemented
21C23×C4×C89C2211yescharacteristic
22C2×M16×C89C22112yescomplemented
23C22×C8×C89C2212yescomplemented
24C23×D89—21no—
25C23×D89—22no—
26C23×D89—212no—
27C23×D89—48no—
28#13083574—224no—
29#13083574C23116yescomplemented
30C2×Dic178—23no—
31C2×Dic178C2316yescomplemented
32C2×D356—26no—
33C2×D356—224no—
34C2×D356—212no—
35C2×D356—23no—
36C2×D356C23112yes—
37C2×D356C23116yescomplemented
38C2×D356C2316yescomplemented
39C2×C22⋊D89—212no—
40C22×Dic89—21no—
41C2×C4×D89—212no—
42#13083563C23132yescomplemented
43D712C23132yescomplemented
44C24×C89D411yescharacteristic
45C22×C4×C89D411yescharacteristic
46C22×C4×C89D416yes—
47C22×C4×C89C2311yescharacteristic
48C22×C4×C89C2316yes—
49M16×C89C23116yescomplemented
50C2×C8×C89C23112yescomplemented
51C23×C89—24no—
52C23×C89C2×D416yes—
53C23×C89C2×D411yescharacteristic
54C22×D89—26no—
55C22×D89—212no—
56C22×D89—216no—
57C22×D89—448no—
58C22×D89—44no—
59C22×D89—48no—
60C2×C4×C89C2×D416yes—
61C2×C4×C89C2×D413yes—
62C2×C4×C89C2×D4112yes—
63C2×C4×C89C2413yes—
64C2×C4×C89C2414yes—
65C22⋊D89—216no—
66C2×Dic89—26no—
67D356—212no—
68D356—216no—
69D356—26no—
70D356C2418yes—
71D356C2414yescomplemented
72C4×D89—216no—
73Dic178—26no—
74Dic178C2414yescomplemented
75C8×C89C2418yescomplemented
76C22×C89—212no—
77C22×C89AGL(1,ℤ/8)14yescomplemented
78C22×C89C22×D414yes—
79C22×C89C22×D413yes—
80C2×D89—24no—
81C2×D89—28no—
82C2×D89—432no—
83C2×D89—412no—
84C2×D89—424no—
85C4×C89C22×D413yes—
86C4×C89C22×D414yes—
87C4×C89C2511yescharacteristic
88Dic89—24no—
89C2×C89—24no—
90C2×C89C2×AGL(1,ℤ/8)16yescomplemented
91C2×C89C23×D411yescharacteristic
92D89—44no—
93D89—48no—
94C22×AGL(1,ℤ/8)—891nomaximal, Sylow
95C89C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
96C22×#41—891no—
97C23×D4—891no—
98C2×AGL(1,ℤ/8)—8924no—
99C22×SD16—892no—
100C22×D8—892no—
101C22×M16D8911yescharacteristic, complemented
102C25—1781no—
103C2×#41—8912no—
104C2×#41—1786no—
105C22×Q8—891no—
106C22×D4—891no—
107C22×D4—892no—
108C22×D4—8912no—
109C22×D4—1781no—
110C22×D4—1786no—
111C22×D4—1781no—
112C23×C4—1781no—
113C23×C4C2×D8911yescharacteristic
114AGL(1,ℤ/8)—8964no—
115C2×SD16—8924no—
116C2×D8—8924no—
117C2×M16C2×D89112yescomplemented
118C22×C8C2×D8912yescomplemented
119C24—3568no—
120C24—17812no—
121C24—1781no—
122C24—1782no—
123C24D35611yescharacteristic
124#41—8916no—
125#41—17824no—
126C2×Q8—896no—
127C2×Q8—1783no—
128C2×D4—896no—
129C2×D4—8912no—
130C2×D4—8916no—
131C2×D4—17812no—
132C2×D4—17812no—
133C2×D4—1786no—
134C2×D4—17824no—
135C2×D4—1783no—
136C22×C4—1781no—
137C22×C4—17812no—
138C22×C4D35616yes—
139C22×C4D35611yescharacteristic
140C22×C4C22×D8916yes—
141C22×C4C22×D8911yescharacteristic
142SD16—8932no—
143D8—8932no—
144M16C22×D89116yescomplemented
145C2×C8C22×D89112yescomplemented
146C23—24no—
147C23—35648no—
148C23—3568no—
149C23—3564no—
150C23—1786no—
151C23—17816no—
152C23—17812no—
153C23C2×D35616yes—
154C23C2×D35611yescharacteristic
155Q8—894no—
156Q8—1786no—
157D4—898no—
158D4—894no—
159D4—17816no—
160D4—17816no—
161D4—17812no—
162D4—1786no—
163C2×C4—1786no—
164C2×C4—17816no—
165C2×C4C2×D35616yes—
166C2×C4C2×D35613yes—
167C2×C4C2×D356112yes—
168C2×C4C23×D8913yes—
169C2×C4C23×D8914yes—
170C8C23×D8918yescomplemented
171C22—212no—
172C22—35632no—
173C22—35624no—
174C22—35612no—
175C22—1784no—
176C22—1788no—
177C22#1566804014yesdirect product factor
178C22C22×D35614yes—
179C22C22×D35613yes—
180C4—1784no—
181C4C22×D35614yes—
182C4C22×D35613yes—
183C4C24×D8911yescharacteristic
184C2—24no—
185C2—3568no—
186C2—3564no—
187C2C2×#1566804016yesdirect product factor
188C2C23×D35611yescharacteristic
subgroupG/H
centerC23C2×D356
commutator, cosocleC4×C89C25
FrattiniC4C24×D89
FittingC22×M16×C89C2
socleC23×C89C2×D4
hypercenterC22×M16D89
hypocenterC89C22×AGL(1,ℤ/8)
LCSG ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#13083574C2semidirect
2. C23×D356C2semidirect
3. C2×#15668040C2direct (N × H)
4. C22×#13083563C2semidirect
5. C22×D712C2semidirect
6. C22×M16×C89C2semidirect
9. C2×#13083574C22semidirect
10. C22×Dic178C22semidirect
14. C22×D356C22semidirect
15. C22×D356C22semidirect
18. #15668040C22direct (N × H)
19. C2×#13083563C22semidirect
20. C2×D712C22semidirect
22. C2×M16×C89C22semidirect
23. C22×C8×C89C22semidirect
29. #13083574C23semidirect
31. C2×Dic178C23semidirect
37. C2×D356C23semidirect
38. C2×D356C23semidirect
42. #13083563C23semidirect
43. D712C23semidirect
49. M16×C89C23semidirect
50. C2×C8×C89C23semidirect
71. D356C24semidirect
74. Dic178C24semidirect
75. C8×C89C24semidirect
77. C22×C89AGL(1,ℤ/8)semidirect
90. C2×C89C2×AGL(1,ℤ/8)semidirect
95. C89C22×AGL(1,ℤ/8)semidirect
101. C22×M16D89semidirect
117. C2×M16C2×D89semidirect
118. C22×C8C2×D89semidirect
144. M16C22×D89semidirect
145. C2×C8C22×D89semidirect
170. C8C23×D89semidirect
177. C22#15668040direct (N × H)
187. C2C2×#15668040direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#130835742
C23×D3562
C2×#156680402
C22×#130835632
C22×D7122
C22×M16×C892
C22×AGL(1,ℤ/8)89